arXiv research

Finite-volume Ricci solitons with constant-length potential are trivial.

This paper is devoted to the study of properties of Killing vector fields of constant length on Riemannian manifolds. If $\mathfrak{g}$ is a Lie algebra of Killing vector fields on a given Riemannian manifold $(M,g)$, and $X\in \mathfrak{g}$ has constant length on $(M,g)$, then we prove that the linear operator $\opera…

New method constructs potential functions for Kähler-Einstein metrics.

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …

In this paper we present some structural results on the Lie algebras of transitive isometry groups of a general compact homogenous Riemannian manifold with nontrivial Killing vector fields of constant length.

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…

Study on a new type of solitons on specific geometric manifolds.

Proves inequality for submanifolds with constant mean curvature.

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We…

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in $\mathbb{R}^3$ with second fundamental form of constant length must be a generalized cylinder $\mathbb{S}^k \times \mathbb{R}^{2-k}$ for some $k\leq2$. Moreover, we prove a gap theorem for smo…

We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N…

We show that if a compact hypersurface $M \subset \mathbb{R}^{n+1}$, $n \geq3$, admits a non zero Killing vector field $X$ of constant length then $n$ is even and $M$ is diffeomorphic to the unit hypersphere of $\mathbb{R}^{n+1}$. Actually, we show that $M$ is a complex ellipsoid in $\mathbb{C}^{N} = \mathbb{R}^{n+1}$.…

Submission was withdrawn by authors. arXiv:math/0305140

In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing pr…

Study on Ricci-like solitons on specific geometric manifolds, finding properties and conditions.

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,ξ)$ where $M = G/H$ is a Riemannian normal homogeneous space, $G$ is a compact simple Li…

It was shown by Seaman that if a compact, oriented 4-dimensional riemannian manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, its intersection form is definite and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomor…

We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length.

In this paper nontrivial Killing vector fields of constant length and corresponding flows on smooth complete Riemannian manifolds are investigated. It is proved that such a flow on symmetric space is free or induced by a free isometric action of the circle $S^1$. The properties of the set of all points with finite (inf…

A locally conformally Kahler manifold is a Hermitian manifold $(M,I,ω)$ satisfying $dω=θ\wedge ω$, where $θ$ is a closed 1-form, called the Lee form of $M$. It is called pluricanonical if $\nablaθ$ is of Hodge type $(2,0)+(0,2)$, where $\nabla$ is the Levi-Civita connection, and Vaisman if $\nablaθ=0$. We show that a c…

Paper finds first examples of unlinked knots that can't be separated.

Study on Yamabe solitons on specific complex manifolds, focusing on torse-forming vector fields.

We discuss Darboux-Staude type of thread configurations for the ellipsoid similar to Chasles-Graves type of thread configurations for the ellipse. These threads are formed by rectilinear segments, geodesic and line of curvature segments on the considered ellipsoid and with tangents tangent to the given ellipsoid and a …

In 2D, Finsler metrics are Douglas and generalized Berwald if they are Berwald or Randers.

We study $n$ dimensional Riemanniann manifolds with harmonic forms of constant length and first Betti number equal to $n-1$ showing that they are 2-steps nilmanifolds with some special metrics. We also characterise, in terms of properties on the product of harmonic forms, the left invariant metrics among them. This all…

In this paper, we obtain some properties of biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ whose shape operator has complex eigen values with at most five distinct prin…

Study on triharmonic curves in Sol space with constant curvature and torsion.

In this paper, we study Clifford-Wolf translations of Finsler spaces. We first give a characterization of Clifford-Wolf translations of Finsler spaces in terms of Killing vector fields. In particular, we show that there is a natural correspondence between Clifford-Wolf translations and the Killing vector fields of cons…

The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation. The main idea leading to the description of a surface M^2 by a spinor field is the observation that the restriction to M^2 of any parallel spinor phi on R^3 is (with respect to t…

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topologi…

In this note we partially answer a question posed by Colbois, Dryden, and El Soufi. Consider the space of constant-volume Riemannian metrics on a connected manifold M which are invariant under the action of a discrete Lie group G. We show that the first eigenvalue of the Laplacian is not bounded above on this space, pr…

Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged right invariant Killing field of constant length. Each such Riemannian metric p…

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists …

Paper establishes convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.

The paper concerns a simple model of bicycle kinematics: a bicycle is represented by an oriented segment of constant length in n-dimensional space that can move in such a way that the velocity of its rear end is aligned with the segment (the rear wheel is fixed on the bicycle frame). Starting with a closed trajectory o…

New classification for Vaisman manifolds with specific properties.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal poss…

It is well-known that if $ξ$ is a smooth vector field on a given Riemannian manifold $M^n$ then $ξ$ naturally defines a submanifold $ξ(M^n)$ transverse to the fibers of the tangent bundle $TM^n$ with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We sh…

The paper proves mass nonnegativity for certain asymptotically locally flat manifolds.

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A_8, is complete and non-singular on R^8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S^4, and …

The paper explores conditions for certain submanifolds to be cylinders.

In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if t…

Develops potential theory for WZW equation in Kähler potentials space.

The paper examines stability of harmonic and symphonic maps with forms and potentials.

The paper examines stability of subelliptic harmonic maps with potential.